Buckling restrained brace with lightweight construction

ABSTRACT

A buckling restrained brace comprises a core member, core restrainer member sections and a jacket member. The core member has two opposite ends. The core restrainer member sections are configured to be arranged around the core member. The jacket member comprises fiber reinforced polymers configured to be wrapped around the core restrainer member sections and core member to couple the core restrainer member sections to the core member such that the core restrainer member sections and jacket member cooperate to provide greater resistance to buckling of the core member when the brace is subjected to compression. In some implementations, the brace has a weight less than about 50% of a weight of a conventional buckling restrained brace of similar length and having a steel core and mortar-filled tubular core restrainer member of comparable cross-sectional areas, respectively.

CROSS-REFERENCE TO RELATED APPLICATION

The present application claims the benefit of U.S. ProvisionalApplication No. 61/584,066, filed Jan. 6, 2012, which is incorporatedherein by reference.

ACKNOWLEDGMENT OF GOVERNMENT SUPPORT

This invention was made with government support under DTRT06-G-0017awarded by the Department of Transportation. The government has certainrights in the invention.

BACKGROUND

A buckling-restrained brace (BRB) is a structural element designed towithstand cyclic loading in the form of repeated tensile and compressiveforces such as from an earthquake or an explosive blast. BRBs addreinforcement and energy dissipation to steel frame buildings to protectthem from large deformations by yielding in tension and compression,while at the same time resisting failure due to buckling.

Conventional BRBs have a steel core member and a surrounding tubularmember filled with mortar that is designed to resist buckling of thecore member when the core member is subjected to compression loading.Although conventional BRBs are adequate in some situations, it would bedesirable to provide BRBs having the same energy dissipating performancewhile having a lower overall weight, which among other advantages makeshandling and installation easier.

SUMMARY

Described below are embodiments of a buckling restrained brace having alight-weight construction.

In an exemplary embodiment, a buckling restrained brace comprises a coremember, core restrainer member sections, and a jacket member. The coremember has two opposite ends. The core restrainer member sections areconfigured to be arranged around the core member. The jacket membercomprises fiber reinforced polymers configured to be wrapped around thecore restrainer member sections and core member to couple the corerestrainer member sections to the core member. The core restrainermember sections and jacket member cooperate to provide greaterresistance to buckling of the core member when the brace is subjected tocompression.

The brace can have a weight less than 50% of a weight of a conventionalbuckling restrained brace of similar length and having a steel core andmortar-filled tubular core restrainer member of comparablecross-sectional areas, respectively.

The core member can have a cross section defining at least one pair ofopposed spaces configured to receive a respective number of corerestrainer member sections. The core member can have a T-shaped crosssection defining two opposed spaces, wherein each of the two spaces isconfigured to receive one of the core restrainer sections. The coremember can be comprised of two-angled sub-members defining a T-shapewhen positioned adjacent each other. The core member can have across-section defining at least four separated spaces, wherein each ofthe spaces is configured to receive one of the core restrainer sections.The core member can be comprised of four angled sub-members, and theangled sub-members can be arranged such that the vertices thereof areadjacent but spaced apart from each other in a cross section of the coremember.

The core member can be comprised of two T-shaped sub-members arrangedopposite to each other. The core restrainer members can be tubular andhave a rectangular cross section or a circular cross section, and aresometimes referred to herein as “tubes”.

The jacket member can be sized to extend over an intermediate portion ofthe core member between the two opposite ends.

The core member can be formed of a ductile material, such as an aluminumalloy. The core member can comprise at least two core member sectionsand at least one spacer member positioned between the core membersections. The spacer member can be formed of a plastic material or fiberreinforced polymers.

The jacket member can comprise at least one layer of material applied atdifferent angles relative to the core member. In some implementations,two or more layers are used. The jacket member can comprise at least onelayer of material applied at an angle of about 30 degrees relative to anaxis of the core member. The core member can be configured to dissipateseismic energy through substantially reversible cyclic plastic strain.The core member, core restrainer member sections, and jacket member canbe constructed of materials selected to reduce corrosion from exposureto environmental conditions.

The core member, core restrainer sections, and jacket member areconfigured to allow the core and the core restrainer sections totranslate relative to each other under pre-defined loading conditionsimposed on the brace.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1( a) is a perspective view of an implementation of a bucklingrestrained brace having a light weight construction with an intermediateportion of the brace cut away to show the relative positions of variouscomponents.

FIG. 1( b) is an enlarged perspective view of a portion of the brace ofFIG. 1( a).

FIGS. 1( c), 1(d), 1(e) and 1(f) are end elevation views ofrepresentative braces having different core member and core restrainermember configurations.

FIG. 1( g) is a perspective view similar to FIG. 1( a) of anotherimplementation of a buckling restrained brace.

FIG. 1( h) is an enlarged perspective view of a portion of the brace ofFIG. 1( g).

FIGS. 2( a) and 2(b) are side elevation views of a representative braceidentifying various dimensions used in modeling.

FIG. 2( c) is an end view of the brace of FIGS. 2( a) and 2(b) showingbolted connections to gusset plates.

FIG. 3( a) is a set of diagrams showing a single degree of freedommechanical model for modeling the brace.

FIG. 3( b) is a drawing showing another model of the brace.

FIG. 4 is a scatter plot of required restrainer stiffness vs. restrainerlength providing a comparison between analytical and numerical results.

FIG. 5( a) is a drawing showing dimensions for two test coupons.

FIG. 5( b) is a perspective view showing a test apparatus for subjectinga test coupon to a predetermined loading.

FIG. 6 is a graph of stress versus axial strain showing the brace'sresponse to predetermined loading.

FIG. 7 is a graph showing maximum cyclic stress versus a number ofreversals for the brace of FIG. 6.

FIG. 8( a) is a graph of normalized stress versus axial strain based ontesting of another representative coupon.

FIG. 8( b) is a side elevation view of the representative coupon testedin FIG. 8( a).

FIG. 9( a) is a perspective view of a core member showing how it ismodeled using finite element analysis.

FIG. 9( b) is a graph of axial load versus axial displacement for themodel of FIG. 9( a).

FIG. 10( a) is a cross section of a core member at an intermediate pointshowing various dimensions used in modeling brace end moments.

FIG. 10( b) is a diagram illustrating end moments or rotations that thebrace of FIG. 10( a) may experience during severe seismic loading.

FIGS. 11( a) through 11(f) show axial load versus log-displacementrelationships for six groups of simulations.

FIG. 11( g) is a graph of the required restrainer stiffness versus theend moment ratio for two groups of braces.

FIG. 12( a) is a perspective view of a brace showing yielding prior tobuckling.

FIG. 12( b) is a perspective view of the brace of FIG. 12( a) showingits deformed shape after buckling.

FIG. 12( c) is an enlarged view of a portion of the brace of FIGS. 12(a) and 12(b) that has been subjected to buckling showing that a plastichinge is created in the area of junction between its full section andits intermediate section.

FIGS. 13( a) and FIG. 13( b) are graphs of axial load versus axialdisplacement for one prototype.

FIGS. 13( c) and FIG. 13( d) are graphs of axial load versus axialstrain at mid-length for the prototype of FIGS. 13( a) and 13(b).

FIG. 13( e) is a plot of normalized cumulative-displacement versusrestrainer stiffness for the prototype of FIGS. 13( a) and 13(b).

DETAILED DESCRIPTION

Small to medium size concrete or steel buildings constructed accordingto deficient legacy codes constitute a large portion of today's backlogof structures requiring seismic retrofit. A number of retrofit solutionsare available to address these deficient structures. However, manysolutions impose great difficulties for material handling andinstallation of traditional lateral elements such as shear walls orconventional steel braces due to limited access for heavy liftingequipment such as cranes and forklifts Therefore, an ultra-lightweightlateral bracing system is desired that allows for easy manual transport,lifting, erection, and connection of required components to the existingstructure without deconstruction of exterior walls for access. Byminimizing disruption to building occupants, the building may remainpartially viable during construction resulting in decreased cost andthus increasing the feasibility of elective upgrades. Described hereinis a new BRB, having an aluminum core member for seismic forcedissipation, and fiber reinforced polymers (FRP) arranged to coupletubular core restrainer sections to the core, thus accomplishing thegoals of decreased installation weight, increased system compactness andefficient energy dissipation.

Attempts to refine metallic seismic dissipaters originally proposed bySkinner et al. (1975) have recently strayed from the traditional steelcore and mortar-filled steel tube restrainer BRBs developed throughoutthe 1980s and 1990s (Watanabe et al. 1988; Wada et al. 1989; Watanabeand Nakamura 1992; Black et al. 2002; Black et al. 2004). Manyvariations have been presented (Xie 2005), but those termed“lightweight” and constructed of bolted or welded all-steel componentsfor both the core and restrainer are the most numerous (Mazzolani et al.2004; Tremblay et al. 2006; Usami et al. 2008; D'Aniello et al. 2008;D'Aniello et al. 2009; Chao and Chen 2009; Ju et al. 2009; Mazzolani etal. 2009). Competing concepts have been characterized as beneficial dueto decreased installation cost, having replaceable cores, ability to uselow-skilled labor for installation, compact for installation confinedspaces, and use in existing building retrofits.

Aluminum as an industrial material has been around for more than acentury but its incorporation into the primary structural elements ofbuildings has been relatively slow with uses limited to secondarysystems such as curtain walls and auxiliary structures such as awnings,canopies or similar structures. However, attempts to utilize its highductility and absence of cyclic hardening in seismic force dissipatingsystems have begun to appear. Shape memory braces constructed withsuper-elastic aluminum alloys that allow a structure to re-center aftera seismic event with little permanent deformation (Mazzolani et al.2004), replaceable shear links constructed of low yield point aluminuminstalled in concentrically braced frames or special truss moment frames(Rai and Wallace 2000) and replaceable aluminum plate shear panels (Rai2002, Mazzolani et al. 2004, Brando et al. 2009) have all been proposedand tested with moderate success.

FRP has successfully been used in structures since the 1970s and hasbeen commonly employed in applications bonded to concrete or steelmembers requiring strengthening or repair (Zhao and Zhang 2007). Morepertinent applications recently have been developed that increaseductility of steel members. For instance, bonded unidirectional sheetswrapped around special truss moment frame chord members enhanced cyclicresponse of plastic hinge behavior (Ekiz et al. 2004). FRP strips bondedto compression elements of flexural members (Accord and Earls 2006),webs of WT compression members (Harries et al. 2009), and HSS columns(Shaat and Fam 2006, 2007, 2009) have also been reported to delay localbuckling of elements subjected to compression. Although applicationswhere the FRP is not bonded to the substrate that it serves to reinforceare rare, they are emerging as an effective method for precludingcompression buckling. Pilot tests of a single steel angle fit with apultruded FRP square tube and wrapped with GFRP (glass fiber reinforcedpolymer) fabric was experimentally loaded in cyclic push-pull testingand achieved an ultimate compressive strength of 35% of the tensilestrength before global buckling (Dusicka and Wiley 2008). Small-scalemonotonic experiments and finite element modeling of rectangular steelbars surrounded by PVC or mortar blocks and wrapped with CFRP (carbonfiber reinforced polymer) fabric have achieved compression loads up toP_(max)/P_(y)=1.53 (Ekiz and El-Tawil 2008). Experimental full-scalecyclic tests of pinned and semi-fixed end steel angles similarly wrappedwith mortar blocks and CFRP fabric achieved up to a 270% increase inenergy dissipation over bare steel angles and compression loads up toP_(max)/P_(y)=0.90 (El-Tawil and Ekiz 2009).

Described below are developments of high-performance BRBs, andspecifically, a new ultra-lightweight BRB, designed for a typical modelbuilding using analytical models developed from established bucklingtheory and experimental cyclic coupon testing of a candidate 6061-T6511aluminum alloy for development of a calibrated constitutive model andfinite element simulations. Analytical models considered both a singledegree of freedom (SDOF) and an established Euler buckling model whichprovided an initial required restrainer stiffness and strength for agiven axial design force and core length. Monotonic numericalsimulations of a prototype brace were performed to examine the effect ofrestrainer stiffness with two different core reduced section lengths andthree degrees of applied end moment. Cyclic simulations were used toassess if a predictable and reliable cumulative plastic ductility andenergy dissipation was possible at a considerable story drift ratio. Aspart of the parametric investigation, end moment effect due to framedrift was considered by using an upper bound approach which considersplastic hinging of the unrestrained section of core.

The new brace utilizes materials readily available in many sizes andprofiles to allow customization of the core-restrainer configuration asshown in FIG. 1( a). Although not directly a part of research, a reviewof past literature and practicality led to the following considerationsfor development: (1) stock extruded aluminum profiles for the coremembers should lower procurement and fabrication costs; (2) bi-planarsymmetry of the brace cross-section should eliminate potential forglobal buckling in a weak direction; (3) non-tapered core cross sectiondimensions should allow a tight fit to the restrainer tubes withoutshimming; (4) back to back core elements should be continuouslysupported by high modulus FRP spacers to prevent core rippling; (5)sufficient space should be provided at the tip of core elements to allowPoisson expansion; (6) unrestrained sections of the core should besufficiently robust to prevent local or torsional buckling modes; (7)the core should be fabricated without welding to prevent materialembrittlement and fatigue notching; (8) a reduced core section should beused to direct plastic straining to the mid-length of the brace awayfrom the vulnerable unrestrained areas; (9) axial independence betweenthe core and restrainer should be maintained using a frictionlessinterface of grease or other lubricant between the FRP tubes andaluminum; and lastly (10) GFRP should be used to prevent galvanicreaction as is present with CFRP and aluminum.

Brace Geometry and Estimate of Strain Demands

Seismic forces, story drift, axial displacement, frame geometry and endconnections were established within the context of a model buildingbased on the SAC 3-story office building located in Los Angeles, Calif.(FEMA 2000). The building consisted of 9.14 m [30 ft] square bays andmeasured 36.6 m by 54.9 m [120 ft by 180 ft] with a story heighth_(i)=3.96 m [13 ft]. Seismic design criteria were taken from thecurrent edition of the building code as follows: S_(s)=2.15 g,S_(ds)=1.43 g, R=7 and C_(d)=5.5 (ASCE 2005). Two adjacent BRB frames(BRBFs) in an inverted v-brace configuration were centered on each ofthe four perimeter column lines. An equivalent lateral force procedurewith 5% minimum eccentricity was used to determine the seismic baseshear and distribution to the individual stories and frames. A bracedesign force of P_(u)=1070 kN [241 k] at the first level was calculatedusing the assumption of equal tension and compression stiffness of theBRBs.

FIGS. 2( a)-(c) show a definition of the brace geometry with a two-stepcore profile. An end to end core length L_(b)=4.83 m [190 in] wasgenerated using assumed W21×111 beams and W14×176 columns to remainconsistent with previous literature reports on testing of full-scaleBRBFs (Fahnestock et al. 2007). Selection of the brace reduced sectionlength L_(c) was subsequently made by considering axial stiffness of thebrace required to limit the inelastic story drift to 2.5%, the maximumallowed by code for a regular structure (ASCE 2005). Calculation of theelastic story drift ratio D_(ie)/h_(i) for a non-prismatic coreneglecting the contribution of the much stiffer beams and columns waspreviously cited by Tremblay et al. (2006) in Eq. (1) whereγ=L_(c)/L_(b) and η=A₁/A₃, F_(y)=core nominal specified yield strength,E_(c)=core Young's modulus and θ=brace angle with horizontal.

$\begin{matrix}{\frac{D_{ie}}{h_{i}} = {\frac{\varphi \; F_{y}}{E_{c}}\left\lbrack \frac{\gamma + {\eta \left( {1 - \gamma} \right)}}{\sin \mspace{11mu} \theta \mspace{14mu} \cos \mspace{14mu} \theta} \right\rbrack}} & (1)\end{matrix}$

By rearranging Eq. (1) algebraically to solve for γ, Eq. (2) is given.D_(ie)/h_(i)=0.45% was calculated by dividing the inelastic story driftof 2.5% by the deflection amplification factor C_(d). Using thevariables θ=42°, E_(c)=69.6 GPa [10,100 ksi], φ=0.9, F_(y)=241 MPa [35ksi] and η=0.456, γ=0.481 was calculated which represents a 2.31 m [91.4in] long reduced section. The final reduced section length was increasedto 2.44 m [96 in] to give γ=0.5 which is termed the Group B brace.Another geometry was created for the parametric study to examine higherexpected axial strain and stiffness with L_(c)=1.47 m [58 in], or γ=0.3,which is termed the Group A brace.

$\begin{matrix}{\gamma = {\left( {1 - \eta} \right)^{- 1}\left\lbrack {{\left( \frac{D_{ie}}{h_{i}} \right)\frac{E_{c}\; \sin \mspace{11mu} \theta \mspace{11mu} \cos \mspace{14mu} \theta}{\varphi \mspace{11mu} F_{y}}} - \eta} \right\rbrack}} & (2)\end{matrix}$

Table 1 shows all prototype brace dimensions.

TABLE 1 Brace prototype dimensions L_(b) L_(c) L_(c2) L_(c3) L_(r) L_(o)L_(s) L_(tr) A₁ A₂ A₃ m m cm cm m cm cm cm cm² cm² cm² Group [in] [in][in] [in] [in] [in] [in] [in] [in²] [in²] [in²] A 4.83 1.47 119 48.33.40 78.7 25.4 15.2 51.1 78.7 112 [190] [58] [47] [19] [134] [31] [10][6] [7.92] [12.2] [17.4] B 4.83 2.44 71.1 48.3 3.40 33.0 25.4 15.2 51.178.7 112 [190] [96] [28] [19] [134] [13] [10] [6] [7.92] [12.2] [17.4]

Approximation of the average inelastic strain in the two-step core atmaximum story drift is required to determine material strain demand.Using Eq. (3) and the previously defined variables, ε_(c)=3.22% and2.25% was calculated for the Group A and B braces at 2.5% story drift,respectively. These values fall within strain amplitudes reported forprevious BRB tests of 1% to 2% for longer core lengths and 3% to 5% forshorter core lengths (Tremblay et al. 2006). Selection of L_(c) shouldtarget an appropriate inelastic strain suitable for use with establishedcyclic properties of the core material as well as brace stiffnessrequired to meet a target design story drift. Typically, connectiondetails, intermediate section overlap length and axial shorteningrequires γ≦0.5 as a practical limit. In the prototype, γ was maximizedby extension of the unwrapped tubes a distance of L_(s) to prevent localbuckling of the intermediate section while still allowing the fullsection to slide through the restrainer.

$\begin{matrix}{ɛ_{c} = {\frac{C_{d}}{L_{c}}\left\lbrack {{D_{ie}\cos \mspace{11mu} \theta} - \left( \frac{n\; \varphi \; {F_{y}\left( {L_{b} - L_{c}} \right)}}{E_{c}} \right)} \right\rbrack}} & (3)\end{matrix}$

SDOF System Analytical Model

Transverse displacement of the slender core member during bucklingimparts flexural demand on the restrainer through application of a forcewith an unknown distribution function w(x). Effort to resist thisdisplacement was conservatively modeled as a simple span restrainer beampinned at the end of length L_(r) assuming the full section of the corerigidly cantilevers from the firmly bolted gusset plate as shown in FIG.2( c). Force interaction between the core and the restrainer was thenestablished using a SDOF mechanical model with axially inextensibletruss members in which an assumed plastic hinge exists at the mid-lengthas shown in FIG. 3( a). Flexural stiffness of the elastic restrainerserves to prevent transverse bifurcation of the core hence increasingthe critical buckling load P_(cr). The plastic hinge was justified byfirst considering the internal core moment by combining elastic columnEqs. (4) and (5) for a pinned-pinned column and solving for the internalmoment at mid-length M_(int) ^(p) at a given transverse displacementΔ_(t) in Eq. (6). Tangent modulus theory was used to account for corematerial non-linearity by replacing E_(c) with E_(ct).

$\begin{matrix}{M_{int} = {E_{c}I_{c}\overset{''}{y}}} & (4) \\{{y(x)} = {\Delta_{t}{\sin \left( \frac{\pi \; x}{L_{r}} \right)}}} & (5) \\{{M_{int}^{p}\left( \frac{L_{r}}{2} \right)} = {\frac{\pi^{2}\Delta_{t}}{L_{r}^{2}}E_{ct}I_{c}}} & (6)\end{matrix}$

The resisting moment M_(res) provided by the bundled tube restrainer wascalculated from equilibrium on the column half-length shown in FIG. 3 aand Eq. (7) where E_(r) and I_(r) are the Young's modulus and moment ofinertia of the restrainer, respectively. Comparison of M_(int) ^(p) toM_(res) showed approximately two orders of magnitude difference at acommon transverse displacement Δ_(t). For this exercise, the modulus ofelasticity for the pultruded composite tubes was taken as E_(r)=19.3 GPa[2800 ksi] and I, was calculated from four 10.8 cm by 6.35 mm [4.25 inby ¼ in] square tubes acting compositely. Tangent modulus E_(ct) of thecore was taken as 1% of Young's modulus to account for strain hardening.The sharp transition between elastic and plastic behavior negates theneed for an incremental approach accounting for material non-linearity.

$\begin{matrix}{M_{res} = {\frac{48\mspace{11mu} E_{r}I_{r}\Delta_{t}}{L_{r}^{3}}\sqrt{\left( \frac{L_{r}}{2} \right)^{2} - \Delta_{t}^{2}}}} & (7)\end{matrix}$

Effect of the restrainer was simulated by an elastic spring withstiffness k_(s) providing a force F exerted at the mid-length of thecore. The spring stiffness is taken from the elastic deflection of abeam loaded at mid-span as F/Δ_(t)=48E_(r)I_(r)/L_(r) ³. Thisrelationship is substituted for k_(s) in Eq. (8) and gives the criticalbuckling load all in terms of known variables after using the classicaleigenvalue solution for the SDOF system. Eq. (9) solves for the requiredrestrainer stiffness E_(r)I_(r)/L_(r) ³ which can be used for intialrestrainer sizing.

$\begin{matrix}{P_{cr} = {\frac{k_{s}L_{r}}{4} = \frac{12\; E_{r}I_{r}L_{r}}{L_{r}^{3}}}} & (8) \\{\frac{E_{r}I_{r}}{L_{r}^{3}} = \frac{P_{u}}{12\; L_{r}}} & (9)\end{matrix}$

Euler Analytical Model

Previous research by Black et al. (2002) has used the Euler column witha distributed force interaction w(x) between the core and restrainerassuming Hooke's law, geometric perfection, concentric load applicationand small displacements. Introduction of a continuous support is showndiagrammatically in FIG. 3( b) as a pinned-pinned column of length L_(r)supported by an infinite number of axially rigid connector barsconnected to the restrainer also spanning length L_(r). By beginningwith force equilibrium on an arbitrary length of column x andintroducing sinusoidal displacement functions originating from Eq. (5),Eq. (10) was achieved. Eq. (11) arrives from solution of thedifferential equation for the critical buckling load P_(cr) where anundetermined additional factor of safety is proposed by Black (2002) toaccount for geometric imperfections and material non-linearity of thecore. The E_(c)I_(c) term for the core can be omitted due to the muchlower contribution as compared with E_(r)I_(r) as previously explainedand the effective length factor K=1 for the pinned-pinned column.Removal of the E_(c)I_(c) term likewise removes the need for incrementalanalysis considering material non-linearity of the core. Eq. (12)similarly solves for the required restrainer stiffness.

$\begin{matrix}{{{p\frac{^{2}{y(x)}}{x^{2}}} + {E_{c}I_{c}\; \frac{^{4}{y(x)}}{x^{4}}} + {E_{r}I_{r}\frac{^{4}{y(x)}}{x^{4}}}} = 0} & (10) \\{P_{cr} = {\frac{\pi^{2}}{\left( {KL}_{r} \right)^{2}}\left( {{E_{c}I_{c}} + {E_{r}I_{r}}} \right)}} & (11) \\{\frac{E_{r}I_{r}}{L_{r}^{3}} = \frac{P_{u}}{\pi^{2}L_{r}}} & (12)\end{matrix}$

The SDOF and Euler analytical design methods are plotted in FIG. 4 ascontinuous functions illustrating required E_(r)I_(r)/L_(r) ³ forlengths ranging from L_(r)=2 m [78.6 in] to 4.25 m [167 in] and loadsP_(u)=225 kN [50 k] to 1350 kN [300 k]. If no degree of conservatism isprovided, the present prototype brace requires E_(r)I_(r)/L_(r) ³=30.1kN/m [0.172 k/in] and 36.6 kN/m [0.209 k/in] for the SDOF and Eulermethods, respectively. The prototype to be considered in the numericalsimulations, used a restrainer stiffness of 33.4 kN/m [0.191 lain]provided by four 4.25 in×0.25 in bundled tubes. The SDOF model resultedin required stiffness values equal to 82% of the Euler model, indicatingthat it may be unconservative. However, restrainer stiffness wasselected to fall in between these values.

Coupon Testing

Alloy 6061-T6511 is a relatively inexpensive heat-treated structuralaluminum that is available in many extruded profiles that areconformable to square or round FRP tubes. This alloy has proven to bereliable when limited to the elastic range, but reversed cyclic behaviorhas not been reported for Δε_(r)/2≧4% and required investigation todetermine its cyclic behavior such as is reported for plate steels(Dusicka et al. 2007). FIGS. 5( a) and 5(b) define the monotonic tensionand cyclic push-pull coupons machined from 0.875 inch round bar and testsetup which used a MTS load frame with a +/−445 kN [+/−110 k] capacity.The apparatus was manually controlled using a LVDT at a constant strainrate dε/dt for both the monotonic and cyclic tests. Cyclic tests wereperformed with a triangular waveform load history and began with atensile excursion. Individual cyclic tests subjected the coupon toconstant total strain amplitudes Δε_(r)/2=2, 3 and 4% at a cyclic strainratio R_(ε)=−1. Table 2 summarizes experimental results for each of thespecimens.

TABLE 2 6061-T6511 coupon test results Coupon ID Material Property T1 C1C2 C3 C4 f_(0.2), MPa [ksi] 296 [42.9] 297 [43.0] 283 [41.1] 289 [41.9]297 [43.0] f_(u), MPa [ksi] 317 [46.0] 321 [46.5] — — — ε_(y), % 0.380.33 0.481 0.427 0.464 ε_(u), % 21.9 22.3 — — — A_(f), mm² [in²]   53.6[0.0831]   53.1 [0.0823] — — — σ_(true), MPa [ksi] 414 [60.1] 378 [54.8]— — — μ 0.816 0.805 0.852 0.835 0.814 Δε_(t)/2, % — — 2.0 3.0 4.0 dε/dt,× 10³ s⁻¹ 0.05 0.10 0.10 0.10 0.10 2N_(f) — — 48 36 22 Notes: A_(f) =measured cross sectional area at failure surface σ_(true) = truefracture stress (P_(u)/A_(f)) μ = ratio of core nominal specified yieldstrength/experimental yield strength (F_(y)/f_(0.2)) 2N_(f) = number ofreversals to fracture

Since experimental yield strength exceeded nominal specified yieldstrength by approximately 25%, a normalization factor μ=F_(y)/f_(0.2)was introduced. This value was used in creation of the materialconstitutive model in order to remain consistent with the material yieldstrength assumptions made during analytical modeling.

Monotonic results indicated strain hardening equal to 0.89% of Young'smodulus until 5% elongation followed by 0.43% softening until tensilefracture (FIG. 6). Cyclic loops have a bi-asymptotic shape and arecomprised of a linear elastic, smooth non-linear elastic-plastictransition and an approximately linear plastic region (FIG. 7).Transitions between the elastic and plastic regions were less abrupt forthe cyclic tests as compared to the monotonic primarily due to theBauschinger effect. Anomalies at the zero stress level can be attributedto looseness in the double nuts holding the specimens in the fixture.Closeness of the loops indicated that cyclic softening occurred at avery low rate, especially for the 2% test. This is better illustrated inFIG. 7 as a plot of maximum cyclic stress versus number of reversalswhere cyclic softening is negligible after an initial cycle ofhardening. Cyclic softening increased minimally as the strain amplitudewas increased from 2% to 4%. Consequently, isotropic cyclic hardeningwas relatively slow compared to kinematic hardening. This absence ofisotropic hardening has been witnessed in similar tests on 6060-T6aluminum tested to strain amplitudes of 0.4%, 0.8%, and 1.2% (Hopperstadet al. 1995).

Cyclic coupon tests on 6061-T651 alloy have been reported that achieved2N_(f)=142 at Δε_(i)/2=2.5% and R_(ε)=−1 (Brodrick and Spiering 1972).This is up to 3 times greater than was achieved in the present tests atsimilar strain amplitudes. Slight bending in the 4% strain coupon waswitnessed and is manifested in the hysteresis plots by a bend in thelinear portion of the loop beginning at zero stress. Uniaxial stress mayhave been similarly compromised in the 2% and 3% coupons leading topremature fatigue failure due to non-uniform cross section straindistribution. Therefore, use of an hourglass shaped coupon without aprismatic center section is recommended for strain ranges greater than2% to control buckling (ASTM 2004). Use of the current data fordefinition of a cyclic material model is not anticipated tosignificantly affect results since the linear portion of the curve isnot used in its definition. However, suitability of the candidate alloyfor use in high cyclic strain applications requires further experimentalstudy.

Numerical Simulations Cyclic Material Constitutive Model

Representative prediction of cyclic material behavior uses a calibratedgeneral nonlinear combined kinematic-isotropic constituitive model. Thismodel has proven to be capable of simulating the Bauschinger effect,cyclic hardening with plastic shakedown and relaxation of the meanstress (Simulia 2010). Experimental data from coupon C3 normalized byμ_(ave)=0.810 was selected which is in between the approximate inelasticcore strains of 2.25% and 3.22%. The calibration procedure used a 3Dfinite element model of the test coupon comprised of C3D4 tetrahedralcontinuum elements as depicted in FIG. 8( b). Convergence of the finemesh was studied by varying the number of degrees of freedom and theelement polynomial. Coupon simulations were set to run in displacementcontrol for two full cycles to verify calibration. Superposition ofthree backstresses effectively captured the shape of the experimentalhysteresis plots in the Bauschinger region by accounting for strainratcheting effects. Superposition of the experimental and numericalresults for the 2%, 3% and 4% strain amplitudes are illustrated in FIG.8( a) with reasonable correlation. Isotropic hardening was not used dueto its negligible influence on cyclic behavior as shown by the stablemaximum cyclic stress plots.

Finite Element Model Configuration

Numerical models were created using commercially available finiteelement analysis and post-processor software (Simulia 2010) andconfigured as shown in FIG. 9( a). Core angles were modeled as fourseparate 3D planar extrusions meshed with fully integrated, generalpurpose 4-node shell (S4) elements capable of modeling large membranestrains and the restrainer was modeled as a single 1D beam meshed withTimoshenko (B31) elements. Beam element section properties were assignedusing equivalent square tubes representative of the x-x and y-y flexuralstiffness of the bundled restrainer tubes acting compositely. Core andrestrainer nodes were connected with slide-plane connectors to the angletips and slot connectors to the angle vertex to decouple axialinteraction and allow Poisson deformation. The spacing between slottedconnectors was kept at a constant 25.4 mm [1 in] leaving one unsupportednode in between the connector nodes. However, in the present study nolocal buckling imperfections were assigned to promote rippling betweenthe connectors. Boundary conditions were assigned to reference points(RP) positioned at the end of the gusset plate a distance L_(c3) fromthe end of the brace. Load eccentricity for end moments was introducedby offsetting the RP from the centroid a distance e_(l) in the positivey-direction to simulate gusset plate rotation from frame drift andsingle curvature of the brace. The applied end moment was directlyproportional to axial load.

Consideration of thin and thick shell formulations on elastic bucklingbehavior was made by examining load vs. axial displacement behavior ofbraces loaded monotonically to an enforced axial displacement of 50.8 mm[2 in] with adequately and inadequately restrained cores. For thesesimulations, a nominal material yield strength F_(y) was used along witha nominal 1% post-yield hardening. The difference between tension andcompression yield load, buckling load and post-buckling path are shownto be negligible in FIG. 9( b), indicating that transverse shearflexibility is not important for global buckling modes with thickness tocharacteristic length ratios less than 1/15. Model convergence was alsostudied by examining element strains at both the elastic and plasticregions for varying number of degrees of freedom. To stimulate globalbuckling, geometric imperfections were introduced into the mesh from thefirst four buckling modes. Maximum global out-of-straightness ofL_(b)/1000 was assigned for modes 1 & 2 and L_(b)/4000 for modes 3 & 4.

Effect of Brace End Moments

BRBF in-plane drift may introduce end moments or rotations into thebrace during severe seismic loading as shown in FIG. 10( b) for aninverted v-brace configuration. This effect causes additional flexuraldemand on the restrainer above those caused by ideal column bucklingmodels. An upper bound end moment that utilizes the available plasticmoment of the core's intermediate section M_(p)′ was used to quantifythis effect which can be determined by performing a rigorous non-linearpush-over analyses of the BRB/BRBF assembly which is beyond the scope ofthis research. The available plastic moment of the axially loaded memberis reduced below the value of F_(y)Z as is shown diagrammatically inFIG. 10( a) when all four core angles act compositely by shear transferoccurring at the bolted connections. Using this interaction of axialload and moment with an axial load equal to the core nominal yield loadP_(yc)=A₁F_(y), the following values were calculated: d₁=8.43 cm [3.32in], d₂=15.7 mm [6.16 in], Z=215 cm³ [13.1 in³] and M_(p)′=5186 kN-cm[459 k-in]. To express this moment as a ratio of the upper bound, thevariable Ψ=M_(app)/M_(p)′ was introduced where M_(app)=maximum appliedend moment and Ψ<1. Additionally, the relationship ΨM_(p)′ can beconverted to a load-eccentricity relationship for use in numericalsimulations as shown in Eq. (13) where the core nominal yield load isused neglecting the contribution of post-yield hardening and e₁ iscalculated for a desired end moment effect.

$\begin{matrix}{{e\; 1} = \frac{\psi \; M_{p}^{\prime}}{P_{yc}}} & (13)\end{matrix}$

Monotonic Simulations

Parametric numerical simulations of monotonically loaded prototypes werestudied in displacement control for comparison with the proposedanalytical models. Variables used were L_(c), E_(r)I_(r)/L_(r) ³ andΨM_(p)′ as shown in Table 3.

TABLE 3 Numerical simulation parameters and results General ParametersCyclic Results Simulation Simulation Mono/ L_(c) E_(r)I_(r)/L_(r) ³Monotonic Results ΣPΔ_(a) ΣPΔ_(a) Group ID Cyclic m [in] kN/m [k/in] ψP_(e)/P_(yc) P_(max)/P_(yc) Δ_(a)/Δ_(t) kN/m ΣP_(o)Δ_(o) 1A 58-R1-M0 M1.47 [58] 31.5 [0.180] 0 0.859 0.730 0.341 γ = 0.3 58-R2-M0 M 1.47 [58]42.2 [0.241] 0 1.15 0.934 0.362 58-R3-M0 M 1.47 [58] 74.6 [0.426] 0 2.041.43 2.34* 2A 58-R1-M1 M 1.47 [58] 42.2 [0.241] 0.25 1.15 0.908 0.370 γ= 0.3 58-R2-M1 M 1.47 [58] 55.1 [0.315] 0.25 1.50 1.13 0.398 58-R3-M1 M1.47 [58] 83.6 [0.477] 0.25 2.28 1.42 1.78* 3A 58-R1-M2 M 1.47 [58] 55.1[0.315] 0.5 1.50 0.883 0.379 γ = 0.3 58-R2-M2 M 1.47 [58] 70.4 [0.402]0.5 1.92 1.09 0.407 58-R3-M2 M 1.47 [58] 93.2 [0.532] 0.5 2.54 1.391.66* 1B 96-R1-M0 M/C 2.44 [96] 31.5 [0.180] 0 0.859 0.720 0.340 39900.529 γ = 0.5 96-R2-M0 M/C 2.44 [96] 42.2 [0.241] 0 1.15 0.923 0.3625100 0.677 96-R3-M0 M/C 2.44 [96] 66.4 [0.379] 0 1.81 1.29 3.59* 67400.895 2B 96-R1-M1 M/C 2.44 [96] 42.2 [0.241] 0.25 1.15 0.898 0.371 48840.649 γ = 0.5 96-R2-M1 M/C 2.44 [96] 55.1 [0.315] 0.25 1.50 1.12 0.4016019 0.799 96-R3-M1 M/C 2.44 [96] 74,6 [0.426] 0.25 2.04 1.29 2.01* 66530.883 3B 96-R1-M2 M/C 2.44 [96] 42.2 [0.241] 0.5 1.15 0.873 0.379 47000.624 γ = 0.5 96-R2-M2 M/C 2.44 [96] 55.1 [0.315] 0.5 1.50 1.08 0.4095760 0.765 96-R3-M2 M/C 2.44 [96] 83.6 [0.477] 0.5 2.28 1.28 1.82* 66300.880 Notes: P_(e) = restrainer buckling load = π² E_(r)I_(r)/L_(r) ²P_(max) = maximum compressive axial load ΣPΔ_(a) = cumulativeload-displacement ΣP_(o)Δ_(o) = area of ideal trapezoidal hysteresis*indicates successful BRB simulation

Target axial displacement Δ_(bm) relating to 2.5% story drift ascalculated from the model building was multiplied by two as specified bythe cyclic loading protocol for “Qualifying Cyclic Tests ofBuckling-Restrained Braces” (AISC 2005).

FIGS. 11( a)-(f) show axial load vs. log-displacement relationships forthe six groups of simulations. Each dual plot illustrates the ability ofthe trial to meet the target axial displacement before reaching thefailure criteria. The failure criteria were defined as buckling orreaching a limiting transverse displacement at the mid-length of thebrace. Maximum transverse displacement Δ_(t) ^(max)=8.79 cm [3.47 in] isdenoted as a dashed line and was calculated by considering f_(b)=206 MPa[30 ksi] and c=11.8 cm [4.63 in] as measured from the baseline four 10.8cm by 6.35 mm [4.25 in by 0.25 in] tube configuration. The relationshipgiven in Eq. (14) was derived from M_(r)=FL_(r)/4, f_(b)=M_(r)/S_(r),S_(r)=I_(r)/c and Δ_(t)=FL_(r) ³/48E_(r)I_(r) where M_(r) and S_(r) arethe flexural moment when flexurally loaded by a point load at mid-spanand elastic section modulus of the restrainer, respectively.

$\begin{matrix}{\Delta_{t}^{\max} = \frac{f_{b}L_{r}^{2}}{12{cE}_{r}}} & (14)\end{matrix}$

Failure points of inadequately restrained braces are denoted by whitemarkers on the plots while end of simulation points for adequatelyrestrained braces are denoted by black markers. Inadequately restrainedbraces generally exhibited a failure progression in compression as shownin FIGS. 12( a) to 12(c) and described as follows: 1) uniform axialstress and yielding at the reduced section with transverse bending; 2)increasing transverse displacement and bending stress at the ends of therestrainer; 3) plastic local buckling of the core angle legs leading tohinging; and 4) overall global buckling.

Numerical results are given in Table 3 for each simulation. Restrainerstiffness used in the third simulation for each group was determined byan iterative process of increasing E_(r)I_(r)/L_(r) ³ to achieve stableBRB performance with P_(max)/P_(yc)>1 and Δ_(a)/Δ_(t)>1.28 whichrepresents target axial displacement over Δ_(t) ^(max). Application ofend eccentricity had a degradation effect on the Δ_(a)/Δ_(t) ratio, butsuccessful simulations were able to remain in relatively straight axialalignment. FIG. 11( g) shows the linear effect of application of endeccentricity on required E_(r)I_(r)/L_(r) ³. Slope of the lines remainedconstant between the Group A and B braces demonstrating that reducedsection length has insignificant effect. Furthermore, end eccentricitymay be accounted for by superimposing from 34.4 to 37.2 kN/m ofadditional stiffness per unit of Ψ which effectively doubles requiredE_(r)I_(r)/L_(r) ³ for this prototype. FIG. 4 illustrates scatter plotcomparison between analytical and numerical results. Numericalsimulations resulted in approximately two times greater requiredE_(r)I_(r)/L_(r) ³ than analytical for the examined brace lengthindicating that a degree of conservatism of two or greater may berequired to account for material non-linearity, load eccentricity,reasonable transverse displacement as well as possible local bucklingeffects near the end of the restrainer. Although, it is recognized thatfurther study is required to determine the degree of conservatismrequired for other brace lengths since only one length was considered inthis research. Accounting for this larger required stiffness, four 5 inby 5/16 in or 5½ in by 5/16 bundled tubes would be required for Ψ=0 andΨ=0.5, respectively. This is within reasonable practical limits forbrace compactness and promotes the notion that the described brace is aviable concept.

Cyclic Simulations

The objectives included to assess energy dissipation potential andassert numerical model stability and repeatability when subjected tocyclic axial force and rotational demand when subjected to a minimumcumulative inelastic axial deformation of 200 times the yielddeformation (AISC 2005). Numerical formulation used the calibratedcyclic constitutive model and did not include simulation of materialfatigue failure.

Table 3 shows test parameters for the Group B brace. Representativehysteresis plots for Group 1B prototypes for load vs. axial displacementand load vs. transverse displacement are given in FIGS. 13( a)-13(d).Inadequately restrained braces exhibited large transverse displacementalong with pinched hysteresis loops on the compression excursions whileadequately restrained braces exhibited full symmetrical loops withminimal transverse displacement. Group 1B hysteresis plots of load vs.axial strain at the mid-length of the reduced section demonstratetension side strain ratcheting for the inadequately restrained brace(96-R1-M0) and nearly symmetrical loops for the adequately restrainedbrace (96-R3-M0). A strain shift of approximately 2.5% is witnessedtoward the tension side due to incomplete strain reversal duringcompression excursions due to transverse displacement. Average achievedmaterial strain over a gage length of 2.54 cm [1 in] at the mid-lengthof the reduced section was numerically measured at the 1.0Δ_(bm) cycleas +2.89% to −1.72% and +3.20% to −2.52% for the R1 and R3 braces,respectively. This correlates reasonably well with ε_(t)=+/−2.25% ascalculated from Eq. (3) with the caveat that positive tension strainsare approximately 25% greater than compression strains in an adequatelyrestrained brace due to the strain ratchetting.

Cumulative energy dissipation ΣPΔ_(a) and load-strain ΣPε_(t) weredetermined by numerically integrating the area under the curve for eachof the nine simulations. Table 3 shows these results along with thosenormalized by an ideal trapezoidal hysteresis plot ΣP_(o)Δ_(o)=7530 kN-m[66,700 k-in]. Adequately restrained braces achieved nearly 90% of theenergy dissipation of the idealized hysteresis. Inadequately restrainedbraces ranged from 53% to 83% showing a marked improvement. Endeccentricity also had a significant effect on cumulative displacementand strain demand as witnessed in FIG. 13( e) where a steeper slope ispresent at higher values of Ψ up to the plateau of ΣPΔ_(a)=6600 kN/m[58,500 k-in]. ΣPε_(t) plots exhibit a lower slope reaching a plateau of2900 kN-m/m [652 k-in/in].

FRP Wrap Design

Bundled FRP tubes should work compositely to achieve greatest strengthand stiffness. Maximum expected shear flow q_(max) between the tubes wasapproximated by utilizing the same Euler column buckling model and forceequilibrium method. Since failure of the bundled tube assembly should becontrolled by the flexural moment of the tubes and not shear failure ofthe wrap, shear flow was determined using the previously related Δ_(t)^(max). Previously defined values of c, f_(b) and L_(r) were used withthe tube arrangement to calculate q_(max)=8.60 kN/cm [4.91 k/in] at theend of the restrainer. A multi-layer wet layup GFRP uniaxial fabric canresist this shear flow. Proprietary wrap systems are common andtypically exhibit ultimate tensile strengths of 582 MPa [84.4 ksi] inthe primary fiber direction and have an effective laminate thickness of1.27 mm [0.05 in]. A truss-like mechanism was conceived using two layersof wrap along the entire length of the restrainer oriented at +/−30°from the longitudinal axis to resist shear flow in tension through theprimary fibers. The allowable shear strength of the wrap was calculatedas 9.85 kN/cm [5.63 k/in] using a degree of conservatism of 1.5 toaccount for additional extreme fiber longitudinal stress imparted bybending of the restrainer assembly. Although, additional stress in thewrap from bending is expected to be minimal since the modular ratio ofthe fabric and tubes is unity.

Weight Reduction

The described prototype brace was calculated to weigh 200 kg [440 lb] or27% and 41% the weight of a traditional mortar-filled tube and all-steelBRB of similar length, core area and restrainer dimensions. Thus, thedescribed prototype brace weighs less than 50% of a comparableconventional brace. For the mortar-filled tube, a single square tube ofcomparable size (8 in×¼ in) was used. Since the nominal yield strengthof common steel and 6061-T6 aluminum are almost identical, similar coresizes were considered fair comparison. Nominal unit weight for mildsteel, aluminum and concrete mortar were taken as 7860 kg/m³ [490lbs/ft³], 2650 kg/m³ [165 lbs/ft³] and 2410 kg/m³ [150 lbs/ft³],respectively. This comparison serves to highlight the considerableweight savings that can be realized with the described brace.

Notation

A₁ = core reduced cross sectional area A₂ = core intermediate crosssectional area A₃ = core full cross sectional area c = distance from NAto restrainer extreme fiber C_(d) = deflection amplification factorD_(ie) = elastic story drift E_(c) = core Young's modulus E_(ct) = coretangent modulus E_(r) = restrainer Young's modulus f_(0.2) =experimental 0.2% offset yield strength f_(b) = restrainer ultimatebending stress F = transverse restrainer force F_(y) = core nominalspecified yield strength h_(i) = story height, level “i” I_(c) = coremoment of inertia I_(r) = restrainer moment of inertia k_(s) =equivalent restrainer spring stiffness K = effective length factor L_(b)= brace end to end length L_(c) = core reduced section length L_(r) =restrainer length M_(app) = maximum applied end moment M_(int) ^(p) =core internal moment M_(p)′ = available plastic moment of intermediatesection M_(r) = restrainer moment at mid-length M_(res) = restrainerresisting moment P = brace applied axial load P_(cr) = critical bucklingload P_(u) = design axial force P_(yc) = core nominal yield load q_(max)= maximum wrap shear flow R_(ε) = cyclic strain ratio γ = L_(c)/L_(b)Δ_(a) = core axial displacement Δ_(bm) = brace disp. at design storydrift Δ_(t) = restrainer transverse displacement at mid-length Δ_(t)^(max) = max. transverse disp. of restrainer ε_(c) = core inelasticstrain ε_(t) = total experimental strain η = A₁/A₃ θ = brace angle withhorizontal μ = F_(y)/f_(0.2) Ψ = M_(app)/M_(p)′

EXEMPLARY EMBODIMENTS

Referring to FIGS. 1( a) and 1(b), an exemplary embodiment of a bucklingrestrained brace 10, sometime referred to as a “full cruciform” type, isshown. The brace has an elongate core member 12 with opposite ends 14,16. At least one spacer member 18 is positioned on the core member 12.In the illustrated implementation, there is a first spacer member 18oriented along one plane of the core member 12 and a second space memberoriented perpendicular to the first spacer member. There are one or morecore restrainer member sections 20 arranged adjacent the spacer memberand around the core member 12. The core restrainer member sections 20are coupled together with the spacer member 18 and the core member 12 bya jacket 19 comprising fiber reinforced polymer fabric that isconfigured to be wrapped around the assembled core member and corerestrainer member sections with the spacer member sandwichedtherebetween.

In exemplary embodiments, the core member is made of aluminum, althoughother materials with suitable ductility could be used. In theillustrated embodiments, the ends 14, 16 of the core member 12 areexposed.

In the embodiment of FIGS. 1( a) and 1(b), there are four corerestrainer member sections 20, but any suitable number of sections maybe used. The core restrainer member sections 20 can have a rectangular(or square) cross-section as shown in FIGS. 1( a), 1(b), 1(c) and 1(e),a circular cross section as shown for the core restrainer members 24 inFIGS. 1( d) and 1(f), or any other suitable cross-section. Asillustrated, the core restrainer sections may have a hollow tubularconfiguration over at least a portion of their length.

The core member may be comprised of a single member or severalsub-members. In the illustrated implementation, the core member 12 iscomprised of comprised of four angles 22 a, 22 b, 22 c, and 22 darranged such that adjacent side surfaces are in contact with each otherand the vertices are adjacent each other and oriented toward the centeras shown. As shown in FIGS. 1( e) and 1(f), the core member 12 can becomprised of two tee members 26 a, 26 b arranged opposite each other,i.e., with the respective uninterrupted side surfaces facing each other.In some embodiments, the multiple sub-members are separated from eachother, e.g., by the interposed spacer member(s), over at least anintermediate portion of the length of the brace 10. Also, exemplaryconfigurations define at least one pair of separated spaces (such as twopair or four spaces as shown in FIGS. 1( c)-1(f)) for receiving the corerestrainer member sections.

In some embodiments, a debonding material such as PTFE can be appliedbetween adjacent surfaces of the core restrainer member sections 20 andthe core member 12 to ensure that there is no coupling or bondingbetween the adjacent surfaces. In some embodiments, no such debondingmaterial is used.

Referring to FIGS. 1( g) and 1(h), another exemplary embodiment of abuckling restrained brace 210 is shown. The brace 210 is similar to thebrace 10 of FIGS. 1( a) and 1(b), except the brace 210 has an elongatecore member 212 formed in a T-shape (shown inverted in the figures), andthere are two core restrainer member sections 220 received in the spacesdefined on either side of the core member 212. As in the case of thebrace 10, a jacket 219 of fiber reinforced polymer fabric is wrappedaround the core restrainer member sections 220 and the core member 212.

In specific implementations, the core restrainer member sections 20, 200are made of fiber reinforced polymers. The core member 12, 212 is madeof a suitable material, such as, e.g., an aluminum alloy.

In the illustrated embodiment, the brace 210 does not include any spacermember, but one or more spacer members can be provided if desired or ifrequired in certain circumstances.

This analytical and numerical study focusing on global bucklingdemonstrated the ability to develop a new ultra-lightweightbuckling-restrained brace for potential application in existing buildingseismic retrofit situations similar to a representative 3-story officemodel building. Calculated required restrainer stiffness from a newlydeveloped SDOF model and previously established Euler buckling model wascompared with monotonic and cyclic numerical simulations of prototypeswith varying restrainer stiffness, reduced section length and appliedend moment. The following presents a summary of results:

(1) A common structural aluminum coupon was tested cyclically to developa hysteresis for use in creation of a constitutive model for cyclicnumerical modeling. Excellent correlation was illustrated with a generalnonlinear combined hardening model using finite element simulations.Test results indicated that low monotonic strain hardening andnegligible cyclic hardening make 6061-T6511 a potentially suitablecandidate for seismic applications.

(2) An important contribution was made to modeling BRB behavior usingnumerical finite element simulations to verify existing analytical baseddesign methods. Monotonic numerical results indicated that a degree ofconservatism of two or greater was required for the considered bracelength when using proposed analytical methods to account for geometricand material non-linearity, local buckling at the unrestrained core andlimiting transverse bending stress on the restrainer. SDOF and Eulerbuckling models achieved similar results with only 13% difference.Further research was recommended to examine the effect different bracelengths have on the degree of conservatism required to achieve BRBperformance.

(3) BRB applied end moment was quantified using an upper bound approachin lieu of performing specific frame analyses in order to account forstory drift two times greater than the maximum 2.5% given in typicalbuilding codes. The restrainer demand from applied end moment wasdetermined by monotonic simulations to be one of linear superpositionwith conventional buckling demand. Additional required stiffness of 34.4to 37.2 kN/m per unit of Ψ was required for the brace length examined.This relationship was shown to hold for restrainer length ratios ofγ=0.3 and 0.5 indicating that there may be potential for using themethod as a quick and easy design aid to account for end moment effectson BRB performance.

(4) Cyclic simulations indicated that reliable BRB performance wasachieved with approximately 90% efficiency as compared to an idealtrapezoidal hysteresis if adequate restrainer stiffness was provided.Lower values of 53% to 83% were typical with inadequate restraint withmost of the cumulative ductility occurring from yielding in the tensionexcursions. Material strains achieved in the simulations correlatedreasonably well with those estimated by simple analytical methods andwere approximately +3.2% to −2.52% for the design story drift. Materialstrains were shown to be asymmetrical due to strain ratcheting caused bytransverse bending.

In view of the many possible embodiments to which the disclosedprinciples may be applied, it should be recognized that the illustratedembodiments are only preferred examples and should not be taken aslimiting in scope. Rather, the scope of protection is defined by thefollowing claims.

1. A buckling restrained brace, comprising: a core member having twoopposite ends; core restrainer member sections configured to be arrangedaround the core member; a jacket member comprising fiber reinforcedpolymers configured to be wrapped around the core restrainer membersections and core member to couple the core restrainer member sectionsto the core member, wherein the core restrainer member sections andjacket member cooperate to provide greater resistance to buckling of thecore member when the brace is subjected to compression.
 2. The bucklingrestrained brace of claim 1, wherein the brace has a weight less thanabout 50% of a weight of a conventional buckling restrained brace ofsimilar length and having a steel core and mortar-filled tubular corerestrainer member of comparable cross-sectional areas, respectively. 3.The buckling restrained brace of claim 1, wherein the core member has across section defining at least one pair of opposed spaces configured toreceive a respective number of core restrainer member sections.
 4. Thebuckling restrained brace of claim 1, wherein the core member has aT-shaped cross section defining two opposed spaces, wherein each of thetwo spaces is configured to receive one of the core restrainer sections.5. The buckling restrained brace of claim 4, wherein the core member iscomprised of two angled sub-members defining a T-shape when positionedadjacent each other.
 6. The buckling restrained brace of claim 1,wherein the core member has a cross section defining at least fourseparated spaces, wherein each of the spaces is configured to receiveone of the core restrainer sections.
 7. The buckling restrained brace ofclaim 1, wherein the core member is comprised of four angledsub-members, and the angled sub-members are arranged such that thevertices are adjacent each other in a cross section of the core member.8. The buckling restrained brace of claim 1, wherein the core member iscomprised of two T-shaped sub-members, and the T-shaped sub-members arearranged opposite to each other in a cross section of the core member.9. The buckling restrained brace of claim 1, wherein the core restrainermembers are tubular and have a rectangular cross section.
 10. Thebuckling restrained brace of claim 1, wherein the core restrainermembers are tubular and have a circular cross section.
 11. The bucklingrestrained brace of claim 1, wherein the jacket member is sized toextend over an intermediate portion of the core member between the twoopposite ends.
 12. The buckling restrained brace of claim 1, wherein thecore member is formed of a ductile material.
 13. The buckling restrainedbrace of claim 1, wherein the core member is formed of an aluminumalloy.
 14. The buckling restrained brace of claim 1, wherein the coremember comprises at least two core member sections, further comprisingat least one spacer member positioned between the core member sections.15. The buckling restrained brace of claim 14, wherein the at least onespacer member is formed of at least one of a plastic material and fiberreinforced polymers.
 16. The buckling restrained brace of claim 1,wherein the jacket member comprises at least one layer of fabricmaterial applied at different angles relative to the core member. 17.The buckling restrained brace of claim 1, wherein the jacket membercomprises at least one layer of fabric material applied at an angle ofabout 30 degrees relative to an axis of the core member.
 18. Thebuckling restrained brace of claim 1, wherein the core member isconfigured to dissipate seismic energy through substantially reversiblecyclic plastic strain.
 19. The buckling restrained brace of claim 1,wherein the core member, core restrainer member sections, and jacketmember are constructed of materials selected to reduce corrosion fromexposure to environmental conditions.
 20. The buckling restrained braceof claim 1, wherein the core member, core restrainer sections and jacketmember are configured to allow the core and the core restrainer sectionsto translate relative to each other under predefined loading conditionsimposed on the brace.